GANs: Should Generator update weights when Discriminator says false continuously
In general, you should train both discriminator D and generator G simultaneously.

Depending on the metric that you use as the target for your model, you may encounter a Vanishing gradient problem. It can happen when you implement original loss (i.e. JS-divergence). In that case D can become overconfident regarding fake samples and won't provide any useful feedback to the G. To find out if training fell into this problem, you should plot D and G loss. It will look as follows:



The original GAN has a lot of problems, that's why I suggest you to use Wasserstein metric. More information you can find in WGAN paper

Here you can find more information about GAN problems:


https://arxiv.org/pdf/1904.08994.pdf
https://arxiv.org/pdf/1804.00140.pdf